Question

Lifetime of electronics: In a simple random sample of 100 electronic components produced by a certain method, the mean lifetime was 125 hours. Assume that component lifetimes are normally distributed with population standard deviation σ = 20 hours.

Construct a 99.5% confidence interval for the mean battery life. Round the answer to the nearest whole number.

Answer 1

To construct a 99.5% confidence interval for the mean battery life, we use the formula: Confidence interval = sample mean ± z * (σ/√n), where the sample mean, z-score for a 99.5% confidence level, standard deviation, and sample size are plugged in. The rounded confidence interval for the mean battery life is approximately 119 to 131 hours.

Step-by-step explanation:

To construct a 99.5% confidence interval for the mean battery life, we can use the formula:

Confidence interval = sample mean ± z * (σ/√n)

Where:

• Sample mean = 125 hours
• Z-score for a 99.5% confidence level = 2.807
• Standard deviation (σ) = 20 hours
• Sample size (n) = 100

Plugging in the values, we get:

Confidence interval = 125 ± 2.807 * (20/√100)

Confidence interval = 125 ± 2.807 * 2

Confidence interval ≈ 125 ± 5.614

Rounding to the nearest whole number, the confidence interval for the mean battery life is approximately 119 to 131 hours.

Answer 2

The 99.5% confidence interval for the mean battery life is between 119 and 131 hours.

Step-by-step explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.9995)/(2) = 0.0025`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.0025 = 0.9975`, so
`z = 2.81`

Now, find the margin of error M as such

`M = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

`M = 2.81*(20)/(√(100)) = 6`

The lower end of the interval is the sample mean subtracted by M. So it is 125 - 6 = 119 hours

The upper end of the interval is the sample mean added to M. So it is 125 + 6 = 131 hours.

The 99.5% confidence interval for the mean battery life is between 119 and 131 hours.